Transverse electric polarization ($\mathbf{E}=E_y\hat{\mathbf{y}}$)

Relative to a propagation plane, transverse electric polarization has its electric field perpendicular to that plane. For a conducting corrugation independent of the perpendicular coordinate, the scalar electric component obeys the Helmholtz equation with Dirichlet data at the metal surface.

With permittivity and permeability depending only on $z$ and time dependence $e^{i\omega t}$, a transverse electric polarization $\mathbf{E}=\hat{\mathbf{y}}{E}_y(x,z)$ obeys the displayed equation. Maxwell's equations give $H_x=E_{y,z}/(i\omega \mu )$ and $H_z=-E_{y,x}/(i\omega \mu )$. The latter satisfies $H_{z,xx}+H_{z,zz}+({\mu }^{\prime }/\mu )H_{z,z}+[{\omega }^2\mu \epsilon +(\log \mu {)}^{\prime \prime }]H_z=0$. At normal incidence $H_z=0$; the nonzero magnetic component is $H_x$.